I don't understand why Principle of Indifference doesn't resolve Bertrand's Paradox

I understand the points you're making, but I have a few questions I can't figure out. I'd say you probably have a better grasp on this than I do, so if you could help me understand these points I'd honest really appreciate it. I'm glad you're taking the time to discuss this, I think it's very interesting.

The midpoint of a chord is always orthogonal to the radius drawn through it. If you picked a point on a radius and drew a chord crossing it at an angle other than 90 degrees, that point wouldn't be the midpoint of the chord.

Right, but it would be a unique chord. If we pick a radius, and an angle to that radius, we get a unique chord. Two of the three methods use 90 degrees all the time, when any angle would achieve the same result of always picking a unique chord. Using the midpoint is nice because it makes it easier to think about, the logic of why it's unique is obvious. But it doesn't seem like we would have to use 90 degrees.

I guess I'm wondering is there any logic as to why 90 degrees is best? It gives a unique chord when we use it, but so do an infinite number of other angles, so why not use something else?

Selecting any other angle and using that all the time would give a random pick that's equivalenting "fair" to these methods. Or we could also make another random pick over an assumed uniform distribution of angles each time we need narrow down a bunch of chords crossing a radius to just one, and that would seem equally "fair" too?

I think this would mean we don't just have methods 2 and 3, but an infinite number of methods for each one where we pick some other angle consistently? Plus another couple answers where we randomly select an angle each time we need one?

To me this reinforces what Bertrand was saying, that you need to be careful when applying the Principle of Indifference and assuming it's OK to use a uniform distribution with an infinite probability distribution. But then I also have to wonder if we're just applying the Principle of Indifference much too liberally. What should count as "evidence" that we shouldn't be assuming a uniform distribution? When we get an infinite number of contradictory answers by applying the Principle of Indifference, is that "evidence" we're using it wrong?

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